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Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…

Numerical Analysis · Mathematics 2025-07-10 Georgios Grekas , Charalambos G. Makridakis , Tristan Pryer

Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to…

Numerical Analysis · Mathematics 2024-11-20 Tim De Ryck , Siddhartha Mishra

Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the residual of the PDE into the loss function of a…

Optimization and Control · Mathematics 2022-11-07 Saviz Mowlavi , Saleh Nabi

In complex engineering systems such as electro-thermal-fluid coupling, rapid and accurate prediction of multi-physics fields is essential for advanced applications like digital twins and real-time condition monitoring. Traditional numerical…

Computational Physics · Physics 2026-03-25 Baitong Zhou , Ze Tao , Fujun Liu , Xuan Fang

In recent years, Physics-Informed Neural Networks (PINNs) have become a representative method for solving partial differential equations (PDEs) with neural networks. PINNs provide a novel approach to solving PDEs through optimization…

Computational Physics · Physics 2024-11-28 Weiwei Zhang , Wei Suo , Jiahao Song , Wenbo Cao

Physics-informed neural networks (PINNs) show great advantages in solving partial differential equations. In this paper, we for the first time propose to study conformable time fractional diffusion equations by using PINNs. By solving the…

Numerical Analysis · Mathematics 2021-08-18 Yinlin Ye , Yajing Li , Hongtao Fan , Xinyi Liu , Hongbing Zhang

Numerical modeling errors are unavoidable in finite element analysis. The presence of model errors inherently reflects both model accuracy and uncertainty. To date there have been few methods for explicitly quantifying errors at points of…

Machine Learning · Computer Science 2024-11-19 Bozhou Zhuang , Sashank Rana , Brandon Jones , Danny Smyl

Physics-informed neural networks (PINNs) have recently been used to solve various computational problems which are governed by partial differential equations (PDEs). In this paper, we propose a multi-output physics-informed neural network…

Computational Engineering, Finance, and Science · Computer Science 2022-12-07 Mingyuan Yang , John T. Foster

Physics-informed neural networks (PINNs) have proven to be a promising method for the rapid solving of partial differential equations (PDEs) in both forward and inverse problems. However, due to the smoothness assumption of functions…

Computational Physics · Physics 2026-03-25 Guoqiang Lei , D. Exposito , Xuerui Mao

Physically informed neural networks (PINNs) are a promising emerging method for solving differential equations. As in many other deep learning approaches, the choice of PINN design and training protocol requires careful craftsmanship. Here,…

Machine Learning · Statistics 2023-10-09 Inbar Seroussi , Asaf Miron , Zohar Ringel

Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not…

Numerical Analysis · Mathematics 2026-03-09 Saad Qadeer , Panos Stinis

Physics-informed neural networks (PINNs) have proven a suitable mathematical scaffold for solving inverse ordinary (ODE) and partial differential equations (PDE). Typical inverse PINNs are formulated as soft-constrained multi-objective…

Machine Learning · Computer Science 2023-04-17 Gabriel S. Gusmão , Andrew J. Medford

Solving differential equations efficiently and accurately sits at the heart of progress in many areas of scientific research, from classical dynamical systems to quantum mechanics. There is a surge of interest in using Physics-Informed…

Machine Learning · Computer Science 2022-07-06 Shaan Desai , Marios Mattheakis , Hayden Joy , Pavlos Protopapas , Stephen Roberts

Physics-informed neural networks (PINNs) are numerical solvers that embed all the physical information of a system into the loss function of a neural network. In this way the learned solution accounts for data (if available), the governing…

Computational Physics · Physics 2025-07-30 Andrés Martínez-Esteban , Pablo Calvo-Barlés , Luis Martín-Moreno , Sergio G Rodrigo

Physics-informed neural networks (PINNs) have been proposed to solve two main classes of problems: data-driven solutions and data-driven discovery of partial differential equations. This task becomes prohibitive when such data is highly…

Machine Learning · Computer Science 2022-10-20 Wei Peng , Wen Yao , Weien Zhou , Xiaoya Zhang , Weijie Yao

Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to…

Numerical Analysis · Mathematics 2023-05-23 Victorita Dolean , Alexander Heinlein , Siddhartha Mishra , Ben Moseley

The research in Artificial Intelligence methods with potential applications in science has become an essential task in the scientific community last years. Physics Informed Neural Networks (PINNs) is one of this methods and represent a…

Computational Physics · Physics 2023-07-24 Luis Medrano Navarro , Luis Martín Moreno , Sergio G Rodrigo

Physics-Informed Neural Networks (PINNs) offer a promising approach to solving differential equations and, more generally, to applying deep learning to problems in the physical sciences. We adopt a recently developed transfer learning…

Machine Learning · Computer Science 2022-11-02 Raphaël Pellegrin , Blake Bullwinkel , Marios Mattheakis , Pavlos Protopapas

Solving time-dependent partial differential equations (PDEs) is fundamental to modeling critical phenomena across science and engineering. Physics-Informed Neural Networks (PINNs) solve PDEs using deep learning. However, PINNs perform…

Machine Learning · Computer Science 2025-08-25 Mayank Nagda , Jephte Abijuru , Phil Ostheimer , Marius Kloft , Sophie Fellenz

Physics-Informed Neural Networks (PINNs) provide a framework for integrating physical laws with data. However, their application to Prognostics and Health Management (PHM) remains constrained by the limited uncertainty quantification (UQ)…

Machine Learning · Computer Science 2026-01-08 Ibai Ramirez , Jokin Alcibar , Joel Pino , Mikel Sanz , Jose I. Aizpurua